When the Stone Lets Water Through

Take a big square grid and a single dial, a probability p between 0 and 1. Go cell by cell and, independently, paint each one open with probability p and leave it closed otherwise. Now look only at the open cells, and call two of them connected if they touch edge-to-edge. They fall into clusters — blobs of mutually-reachable open cells.

When p is small, the open cells are sparse and the clusters are tiny, scattered islands. When p is close to 1, almost everything is open and there is essentially one enormous cluster covering the plane. Somewhere in between, the picture changes — and not gradually. Below a sharp critical value p_c every cluster is finite, with probability one; above it, an infinite cluster appears and spans the whole grid. There is no soft ramp between the two regimes. That abrupt birth of a system-spanning cluster, at a precise point, is the percolation phase transition — the simplest model in which a local, random, independent rule produces a sudden global change of state.

The name and the model are due to S. R. Broadbent and J. M. Hammersley in 1957. Broadbent, an engineer, had been working on gas-mask filters packed with porous carbon granules; he wanted to know when a gas could find a connected path through the random maze of pores. He brought the puzzle to Hammersley, a statistician, and they wrote down the lattice version below — coining percolation, after fluid percolating through a porous medium. Drag the dial and find the threshold yourself.

What you just felt — the clusters refusing to span, refusing, refusing, then abruptly spanning over a tiny range of p — sharpens into a true discontinuity as the grid grows. On an infinite lattice the fraction of cells in the giant cluster is exactly zero below p_c and lifts off the axis above it. That is the order parameter on the right, and the whole game of percolation theory is: where, exactly, is p_c?

Here the model does something humbling. The answer depends on the lattice and on whether you randomize the cells (site percolation) or the connections between them (bond percolation) — and for some of these the threshold is a clean number we can prove, while for others, equally natural, it is a number nobody has ever managed to write down.

Measuring a number we can't write down

You can estimate any threshold by brute force: drop a probability, generate a random grid, ask whether it spans, repeat thousands of times, and read off the p at which spanning crosses fifty-fifty. As the grid grows the spanning curve steepens toward a vertical step at p_c — finite-size scaling, sharpening before your eyes. The instrument below runs that experiment continuously and refines its estimate as the samples pile up.

The needle for bond percolation on the square lattice settles onto one-half, and that is not an accident of the simulation — it is a theorem. The square lattice is self-dual: thread a second lattice through the holes of the first, joining the centres of adjacent faces, and you get another square lattice, congruent to the original. An open path on one lattice is blocked by a closed path on its dual, so the two thresholds must add to one — p_c + p_c* = 1 — which forces the only candidate to be ½. That duality is a heuristic; it tells you ½ is the only possible answer but not that the answer exists there. The rigour came in two steps: Harris (1960) proved there is no spanning cluster at p = ½ (so p_c ≥ ½), and Kesten (1980) closed the gap (p_c ≤ ½). Exactly one-half, proved.

Now slide the same instrument to site percolation on the square lattice — randomize the cells instead of the bonds, the most elementary version of the whole subject. The needle lands near 0.5927. And there the mathematics simply stops. No closed form for this number is known. The best anyone can do is compute it numerically, ever more precisely — currently 0.59274605079210, to a dozen digits (Jacobsen 2015) — with no formula, no exact expression, and no proof that one exists. It is one of the most-computed constants in statistical physics and one of the least understood. Switch to site percolation on the triangular lattice and, remarkably, the threshold snaps back to an exact ½ — provable, because that lattice has a self-matching symmetry the square site lattice lacks. Two exact halves on either side of a number with no name.

The same machine measures all three the same way. Mathematics could prove only two of them. That gap is the honest edge of the field.

Making one-half visible

The duality argument isn't only an asymptotic story. At p = ½ on the square lattice it produces an exact, finite identity you can watch land. Take a rectangle of sites that is one column wider than it is tall — an (n+1) × n box — and open each bond with probability ½. Ask: is there an open path crossing from the left edge to the right edge? Either there is, or there is a closed path on the dual blocking it from top to bottom — and by the lattice's self-duality plus the open↔closed symmetry at p = ½, these two events are mirror images of equal probability. They are also exhaustive and exclusive: exactly one occurs. So the left-right open crossing happens with probability exactly ½, for every n. Not in a limit — exactly, on a finite grid.

The shape of the unknown

Why can we prove ½ for the bonds and not for the sites? Because the bond square lattice hands us a perfect self-duality, and the triangular site lattice a perfect self-matching, and the square site lattice hands us neither. Its threshold is a real number that the model is fully entitled to have — it just appears not to be expressible in any closed form we know how to write, the way the bond answer is. Numerically it is pinned to a dozen digits; algebraically it is a blank.

The deeper structure is not blank, though — it is one of the triumphs of modern probability, and here too the honest line runs right through the middle of it. At the critical point the spanning cluster is a fractal, with dimension 91/48 ≈ 1.896; the way clusters grow as you approach p_c is governed by critical exponents — the correlation length diverges like |p − p_c|^−4/3, the giant cluster lifts off like (p − p_c)^5/36 — numbers first conjectured from the Coulomb gas and conformal field theory (den Nijs, Nienhuis). And the crossing probability you measured in Instrument II has, in the scaling limit, a universal closed form independent of the lattice — Cardy's formula (1992), a hypergeometric expression in the rectangle's shape that Lennart Carleson recast as a startlingly simple linear law on an equilateral triangle.

Here is the catch, and it is the same catch as the threshold. All of that — conformal invariance, Cardy's formula, the exponents, the 91/48 — is proved for exactly one model: site percolation on the triangular lattice, by Stanislav Smirnov in 2001, the work that connected percolation to the Schramm–Loewner Evolution (SLE₆; Schramm 2000) and helped earn Fields Medals for Wendelin Werner (2006) and Smirnov (2010). For the square lattice — bonds and sites alike — every one of these beautiful results remains a conjecture. We believe them, because percolation is expected to be universal: the large-scale critical behaviour shouldn't care which lattice you drew. But "shouldn't care" is not a proof. The exact threshold and the exact crossing law for the square site lattice are, as of today, both genuinely open.